Ph.D. Katarína Bellová Page 1 Mathematics 2 (10-PHY-BIPMA2) EXAM - Solutions, 20 July 2017, 10:00 12:00 All answers to be justified.
|
|
- Carol Gibson
- 6 years ago
- Views:
Transcription
1 PhD Katarína Bellová Page 1 Mathematics 2 (10-PHY-BIPMA2 EXAM - Solutions, 20 July 2017, 10:00 12:00 All answers to be justified Problem 1 [ points]: For which parameters λ R does the following system of equations have a unique solution? x 1 + x 3 x = 2 x 2 2x 3 = 3 x 1 + λx 3 = x 3 x = 2 Solution: A system of linear equations for variables has a unique solution if and only if the correspondent matrix of coefficients has full rank, i e rank Elementary row operations do not change the rank: λ λ λ λ (first we swapped the last two rows, then substracted a multiple of the first and third row resp from the last row The last matrix has rank if and only if λ 0 Problem 2 [ points]: Prove or give a counterexample: For any two 2-by-2 matrices A, B R 2 2, it holds that Det (A + B = Det (A + Det (B Solution: The equality is false( in general, as can( be easily seen by the choice of almost any matrices E g taking A = and B = yields Det (A + B = Det ( 1 0 = 1 0 = Det 0 1 Problem 3 [8 points]: Consider the matrix ( 2 1 A = 1 ( Det 0 0 ( (a Find the eigenvalues and the corresponding eigenvectors of A (b Is A diagonalizable? (c Find the Jordan decomposition A = P 1 JP Write all matrices in the expression explicitely Solution:
2 PhD Katarína Bellová Page 2 Mathematics 2 (10-PHY-BIPMA2 (a Eigenvalues can be characterized as roots of the characteristic polynomial: ( 2 λ 1 Det (A λi 2 = Det = (2 λ( λ + 1 = λ 2 6λ + 9 = (λ λ Hence λ = 3 is the only eigenvalue of A with algebraic multiplicity 2 Corresponding eigenvectors v = (v 1, v 2 solve (A λi 2 v = 0, where ( ( A λi 2 = = Hence, the only eigenvectors are the non-zero solutions of the corresponding homogeneous system of linear equations, which are all non-zero scalar multiples of (1, 1 We also see that the geometric multiplicity of λ = 3 is equal to 1 (as the dimension of the eigenspace (b Since the algebraic multiplicity of λ = 3 is different from its geometric multiplicity (2 1, matrix A is not diagonalizable (c The only possible Jordan form corresponding to a 2 2 matrix with a single eigenvalue λ = 3 with algebraic multiplicity 2 and geometric multiplicity 1 is the matrix ( The transition matrix P 1 (according to the somewhat suboptimal notation from the question can be chosen from any corresponding generalized eigenvectors: first column can be chosen for example as v 1 = (1, 1 t and the second column as any solution v 2 of the system (A λi v 2 = v 1 : Hence we can choose e g v 2 = (0, 1 t and 1v v 2 2 = 1, 1v v 2 2 = 1 P 1 = Computing the inverse as ( ( 1 0 gives P = and finally 1 1 ( 1 0 A = 1 1 ( ( ( ( Problem [6 points]: Let V be an inner product space over a field F (F = R or F = C
3 PhD Katarína Bellová Page 3 Mathematics 2 (10-PHY-BIPMA2 (a For a nonempty subset S V, define the orthogonal complement S (b Let U 1, U 2 be two vector subspaces of V Prove that (U 1 + U 2 = U 1 U 2 Solution: (a The orthogonal complement of S consists of all vectors from V which are orthogonal to all vectors from S: S = {v V : s, v = 0 s S} (b We will show two inclusions and to show the equality of the two sets Choose any vector v (U 1 + U 2, i e v is orthogonal to all vectors from U 1 + U 2 Since U 1 U 1 + U 2 ( 0 U 2, so any u U 1 and be written as u = u + 0, v is in particuar orthogonal to all vectors from U 1, i e v U1 In the same way, U 2 U 1 +U 2, so v U2 Hence v U1 U2 Since v (U 1 +U 2 was arbitrary, we showed (U 1 +U 2 U1 U2 On the other hand, choose any v U1 U2 Then v is orthogonal to all vectors from U 1 as well as all vectors from U 2 To prove that v (U 1 + U 2, we need to show that v is orthogonal to all vectors from U 1 + U 2 Let u (U 1 + U 2 Then u can be written as u = u 1 + u 2, where u 1 U 1, u 2 U 2 Using the linearity of the scalar product and v U1, v U2, we get u, v = u 1 + u 2, v = u 1, v + u 2, v = = 0 Since u (U 1 + U 2 was arbitrary, we see that v (U 1 + U 2 Since v U 1 U 2 was also arbitrary, we get the other inclusion U 1 U 2 (U 1 + U 2 Problem [ points]: Let (e 1, e 2 be an orthonormal basis of a vector space V over C Let T L(V, V be a linear operator such that T (e 1 = e 1 + e 2, T (e 2 = e 1 + e 2 (a Is T self-adjoint? (b Is T unitary? Solution: Matrix of T in basis (e 1, e 2 is A T = ( Since (e 1, e 2 is an orthonormal basis, matrix of the adjoint operator T is equal to A T = A t T = ( ( = (a Operator S is self-adjoint if and only if S = S Since for our T, A T operator is not self-adjoint A T, the
4 PhD Katarína Bellová Page Mathematics 2 (10-PHY-BIPMA2 (b Operator S is unitary if and only if S = S 1, or if and only if SS = id (in finitedimensional space Using the matrices of the operators, this is equvalent to A S A S = I Since indeed ( ( ( 1 0 =, 0 1 operator T is unitary Problem 6 [ points]: Consider the bilinear form B on R 2 R 2 whose Gram matrix in the basis ((1, 0, (0, 1 is ( 1 0 A B = 3 What is the Gram matrix of B in the basis ((2, 1, (1, 1? Solution: Transision matrix from ((1, 0, (0, 1 to ((2, 1, (1, 1 is equal to ( 2 1 P = 1 1 Then matrix of B in basis ((2, 1, (1, 1 is equal to ( ( ( Ã B = P t A B P = = Alternatively, one can compute the terms (ÃB ij as where ẽ i = (2, 1 t, ẽ j = (1, 1 (ÃB ij = B(ẽ i, ẽ j = ẽ t ia B ẽ j, ( Problem 7 [7 points]: Consider the following function f : R 2 R: { xy if (x, y (0, 0, x f(x, y = +y 0 if (x, y = (0, 0 (a Is f continuous on R 2? (b Do the partial derivatives f x, f y exist in the point (0, 0? (c Is f differentiable on R 2? Solution: (a If f was continuous at (0, 0, then we would have lim (x,y (0,0 xy x + y = lim f(x, y = f(0 = 0 (x,y (0,0 In particular, for any continuous curve (x, y = (α(t, β(t with (α(0, β(0 = (0, 0 (and (α(t, β(t (0, 0 for t 0, we would have lim t 0 α(tβ(t α(t + β(t = 0
5 PhD Katarína Bellová Page Mathematics 2 (10-PHY-BIPMA2 However, choosing (α(t, β(t = (t, t leads to lim t 0 α(tβ(t α(t + β(t = lim t 0 t 2 2t = lim t 0 1 2t 2 = +, so f cannot be continuous at 0, and hence is not continuous on R 2 (b Note that on the coordinate axes, f(x, 0 = x 0 = 0 x + 0 (x 0, f(0, y = 0 y = y (y 0 Hence f x (0, 0 = lim x 0 f(x, 0 f(0, 0 x f y (0, 0 = lim y 0 f(0, y f(0, 0 y = lim x x = lim y y = 0, = 0 and we see that both partial derivatives exist at the point (0, 0 (c Since f is not continuous on R 2, it cannot be differentiable on R 2 Problem 8 [ points]: Consider the following function f : R 2 R: f(x, y = cos x e 3x+y2 Compute D (1,2 f(0, 1, i e the directional derivative of f in direction (1, 2 at the point (0, 1 Solution: First compute the gradient: x f(0, 1 = sin xe 3x+y2 + cos xe 3x+y2 3 = 3e, (x,y=(0,1 y f(0, 1 = cos xe 3x+y2 2y = 2e, (x,y=(0,1 so f(0, 1 = (3e, 2e Now note that (1, 2 is not a unit vector Considering a unit vector in the same direction as (1, 2 ( (1, 2 (1, 2 1 l = = (1, = 2,, 2 we get D l (0, 1 = f(0, 1 l = (3e, 2e ( 1, 2 = 3e 1 + 2e 2 = 7e
6 PhD Katarína Bellová Page 6 Mathematics 2 (10-PHY-BIPMA2 Interpreting D (1,2 f(0, 1 as the derivative along the non-unit vector (1, 2 was also considered OK: D (1,2 f(0, 1 = f(0, 1 (1, 2 = 3e 1 + 2e 2 = 7e Problem 9 [ points]: Let w = xyz and consider the spherical coordinates (here r 0, θ [0, π], ϕ [0, 2π r > 0, θ (0, π, ϕ (0, 2π Solution: Using chain rule, x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ Find w ϕ w ϕ = w x x ϕ + w y y ϕ + w z z ϕ = yz r sin θ( sin ϕ + xz r sin θ cos ϕ + xy 0 in terms of r, θ, ϕ in a point where = r sin θ sin ϕ r cos θ r sin θ sin ϕ + r sin θ cos ϕ r cos θ r sin θ cos ϕ = r 3 sin 2 θ cos θ( sin 2 ϕ + cos 2 ϕ = 1 2 r3 sin θ sin(2θ cos(2ϕ Alternatively, one can express w = r 3 sin 2 θ cos θ sin ϕ cos ϕ and differentiate directly Problem 10 [6 points]: Find the maximum and minimum value of the function on the set f(x, y = x 2 + 2xy + 1 {(x, y R 2 : 2x 2 + y 2 9} Solution: First note that the set D = {(x, y R 2 : 2x 2 + y 2 9} is a bounded and closed subset of R 2, and hence the continuous function f(x, y = x 2 + 2xy + 1 attains its maximum and minimum on D The extrema can be attained either inside D = {(x, y R 2 : 2x 2 + y 2 < 9}, or on the boundary D = {(x, y R 2 : 2x 2 + y 2 = 9} Let us find all candidates If (x, y D is a point of extremum, then it is a critical point and we have 0 = f x (x, y = 2x + 2y, 0 = f y (x, y = 2x Second equality yields x = 0, and then first equality yields y = 0 The point (x, y = (0, 0 indeed lies in D and f(0, 0 = = 1 (1 If (x, y D is a point of extremum, we can use the method of Lagrange mulitipliers: setting g(x, y = 2x 2 + y 2 9, there must exist a λ R such that f(x, y = λ g(x, y
7 PhD Katarína Bellová Page 7 Mathematics 2 (10-PHY-BIPMA2 which together with the condition (x, y D gives From (3, we have x = λy Plugging this into (2, we get 2x + 2y = λ x, (2 2x = λ 2y, (3 2x 2 + y 2 = 9 ( 2λy + 2y = λ 2 y, y(2λ 2 λ 1 = 0 If y = 0, then x = λy = 0, violating ( Hence y 0 and necessarily (2λ 2 λ 1 = 0, which has the roots λ = 1 ± { = If λ = 1, then x = λy = y and plugging this into ( yields 3x 2 = 9, x = y = ± 3 (they need to have the same sign In these points, ( ( f 3, 3 = f 3, 3 = = 10 ( In the remaining case λ = 1, we get x = λy = 1 y, and plugging this into ( gives y2 + y2 = 9, yielding y = ± 6 and x = 1y = 6 At these points, 2 2 ( f 6 2, 6 = f ( 6 2, = = 7 2 (6 Comparing the values in (1, ( and (6, we see that f attains its maximum value of 10 at the points ( 3, 3 and ( 3, 3 and its minimum value of 7 at the points 2 ( 6, 6 and ( 6, Problem 11 [ points]: Find the general solution to the differential equation on a maximal possible interval xyy = x 2 + y 2 Solution: The eqeuation is homogeneous, so substitution y(x = xt(x (we can always assume y to have this form for x 0 should lead to an ODE with separated variables Indeed, plugging in y = t + xt yields x 2 t 2 + x 3 tt = x 2 + x 2 t 2, x 3 tt = x 2
8 PhD Katarína Bellová Page 8 Mathematics 2 (10-PHY-BIPMA2 If x 0, the equation can be further written as t dt dx = 1 x, t dt = 1 x dx, 1 2 t2 = ln x + C, t = ± 2(ln x + C, y = ±x 2(ln x + C Given C R, this defines two solutions which are defined for ln x + C 0, i e for x e C or x e C (note that in particular we never get to x = 0 Although the value y(±e C = 0 is finite, we cannot continue the solution up to this point, since it violates the original equation (y = 0, x 0, y does not exist Hence, y(x = ±x 2(ln x + C are solutions on maximal intervals (, e C and (e C, + Note that outside of x 0, these are the only possible solutions (in particular, y(x 0 for x 0 as noted above It remains to look at the case when x = 0 From the original equation we immediately get y(0 = 0 Assume that there is some solution y = y(x on an interval I containing 0 (e g on [0, a with y(0 = 0 As seen above, on interval ( a, a the solution must coincide 2 with some of the above solutions However, this is not possible, since none of the above solutions can be continued upto x = 0 Hence there are no solutions defined at x = 0
SYLLABUS. 1 Linear maps and matrices
Dr. K. Bellová Mathematics 2 (10-PHY-BIPMA2) SYLLABUS 1 Linear maps and matrices Operations with linear maps. Prop 1.1.1: 1) sum, scalar multiple, composition of linear maps are linear maps; 2) L(U, V
More informationLast name: First name: Signature: Student number:
MAT 2141 The final exam Instructor: K. Zaynullin Last name: First name: Signature: Student number: Do not detach the pages of this examination. You may use the back of the pages as scrap paper for calculations,
More information1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det
What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix
More informationReduction to the associated homogeneous system via a particular solution
June PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 5) Linear Algebra This study guide describes briefly the course materials to be covered in MA 5. In order to be qualified for the credit, one
More informationx 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7
Linear Algebra and its Applications-Lab 1 1) Use Gaussian elimination to solve the following systems x 1 + x 2 2x 3 + 4x 4 = 5 1.1) 2x 1 + 2x 2 3x 3 + x 4 = 3 3x 1 + 3x 2 4x 3 2x 4 = 1 x + y + 2z = 4 1.4)
More informationMath 322. Spring 2015 Review Problems for Midterm 2
Linear Algebra: Topic: Linear Independence of vectors. Question. Math 3. Spring Review Problems for Midterm Explain why if A is not square, then either the row vectors or the column vectors of A are linearly
More information235 Final exam review questions
5 Final exam review questions Paul Hacking December 4, 0 () Let A be an n n matrix and T : R n R n, T (x) = Ax the linear transformation with matrix A. What does it mean to say that a vector v R n is an
More informationLinear Algebra Practice Problems
Linear Algebra Practice Problems Page of 7 Linear Algebra Practice Problems These problems cover Chapters 4, 5, 6, and 7 of Elementary Linear Algebra, 6th ed, by Ron Larson and David Falvo (ISBN-3 = 978--68-78376-2,
More informationLINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM
LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator
More information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More informationLinear Algebra problems
Linear Algebra problems 1. Show that the set F = ({1, 0}, +,.) is a field where + and. are defined as 1+1=0, 0+0=0, 0+1=1+0=1, 0.0=0.1=1.0=0, 1.1=1.. Let X be a non-empty set and F be any field. Let X
More informationLinear Algebra: Matrix Eigenvalue Problems
CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given
More informationExam Study Questions for PS10-11 (*=solutions given in the back of the textbook)
Exam Study Questions for PS0- (*=solutions given in the back of the textbook) p 59, Problem p 59 Problem 3 (a)*, 3(b) 3(c) p 55, Problem p547, verify the solutions Eq (8) to the Marcov Processes being
More information1. Select the unique answer (choice) for each problem. Write only the answer.
MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x +
More informationI. Multiple Choice Questions (Answer any eight)
Name of the student : Roll No : CS65: Linear Algebra and Random Processes Exam - Course Instructor : Prashanth L.A. Date : Sep-24, 27 Duration : 5 minutes INSTRUCTIONS: The test will be evaluated ONLY
More informationMATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.
MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. Adjoint operator and adjoint matrix Given a linear operator L on an inner product space V, the adjoint of L is a transformation
More informationMA 265 FINAL EXAM Fall 2012
MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators
More information5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers.
Linear Algebra - Test File - Spring Test # For problems - consider the following system of equations. x + y - z = x + y + 4z = x + y + 6z =.) Solve the system without using your calculator..) Find the
More informationMath 21b. Review for Final Exam
Math 21b. Review for Final Exam Thomas W. Judson Spring 2003 General Information The exam is on Thursday, May 15 from 2:15 am to 5:15 pm in Jefferson 250. Please check with the registrar if you have a
More informationMath 250B Final Exam Review Session Spring 2015 SOLUTIONS
Math 5B Final Exam Review Session Spring 5 SOLUTIONS Problem Solve x x + y + 54te 3t and y x + 4y + 9e 3t λ SOLUTION: We have det(a λi) if and only if if and 4 λ only if λ 3λ This means that the eigenvalues
More information2.3. VECTOR SPACES 25
2.3. VECTOR SPACES 25 2.3 Vector Spaces MATH 294 FALL 982 PRELIM # 3a 2.3. Let C[, ] denote the space of continuous functions defined on the interval [,] (i.e. f(x) is a member of C[, ] if f(x) is continuous
More informationNATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS SEMESTER 2 EXAMINATION, AY 2010/2011. Linear Algebra II. May 2011 Time allowed :
NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS SEMESTER 2 EXAMINATION, AY 2010/2011 Linear Algebra II May 2011 Time allowed : 2 hours INSTRUCTIONS TO CANDIDATES 1. This examination paper contains
More informationNo books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question.
Math 304 Final Exam (May 8) Spring 206 No books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question. Name: Section: Question Points
More informationLINEAR ALGEBRA QUESTION BANK
LINEAR ALGEBRA QUESTION BANK () ( points total) Circle True or False: TRUE / FALSE: If A is any n n matrix, and I n is the n n identity matrix, then I n A = AI n = A. TRUE / FALSE: If A, B are n n matrices,
More information3. Identify and find the general solution of each of the following first order differential equations.
Final Exam MATH 33, Sample Questions. Fall 7. y = Cx 3 3 is the general solution of a differential equation. Find the equation. Answer: y = 3y + 9 xy. y = C x + C x is the general solution of a differential
More informationLINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS
LINEAR ALGEBRA, -I PARTIAL EXAM SOLUTIONS TO PRACTICE PROBLEMS Problem (a) For each of the two matrices below, (i) determine whether it is diagonalizable, (ii) determine whether it is orthogonally diagonalizable,
More informationMAT Linear Algebra Collection of sample exams
MAT 342 - Linear Algebra Collection of sample exams A-x. (0 pts Give the precise definition of the row echelon form. 2. ( 0 pts After performing row reductions on the augmented matrix for a certain system
More informationReview problems for MA 54, Fall 2004.
Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on
More informationMATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL
MATH 3 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL MAIN TOPICS FOR THE FINAL EXAM:. Vectors. Dot product. Cross product. Geometric applications. 2. Row reduction. Null space, column space, row space, left
More informationPRACTICE FINAL EXAM. why. If they are dependent, exhibit a linear dependence relation among them.
Prof A Suciu MTH U37 LINEAR ALGEBRA Spring 2005 PRACTICE FINAL EXAM Are the following vectors independent or dependent? If they are independent, say why If they are dependent, exhibit a linear dependence
More informationAPPLICATIONS The eigenvalues are λ = 5, 5. An orthonormal basis of eigenvectors consists of
CHAPTER III APPLICATIONS The eigenvalues are λ =, An orthonormal basis of eigenvectors consists of, The eigenvalues are λ =, A basis of eigenvectors consists of, 4 which are not perpendicular However,
More informationStudy Guide for Linear Algebra Exam 2
Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real
More informationMath 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination
Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column
More informationEXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. 1. Determinants
EXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. Determinants Ex... Let A = 0 4 4 2 0 and B = 0 3 0. (a) Compute 0 0 0 0 A. (b) Compute det(2a 2 B), det(4a + B), det(2(a 3 B 2 )). 0 t Ex..2. For
More informationExercise Set 7.2. Skills
Orthogonally diagonalizable matrix Spectral decomposition (or eigenvalue decomposition) Schur decomposition Subdiagonal Upper Hessenburg form Upper Hessenburg decomposition Skills Be able to recognize
More informationMATH 240 Spring, Chapter 1: Linear Equations and Matrices
MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear
More informationPractice Final Exam Solutions for Calculus II, Math 1502, December 5, 2013
Practice Final Exam Solutions for Calculus II, Math 5, December 5, 3 Name: Section: Name of TA: This test is to be taken without calculators and notes of any sorts. The allowed time is hours and 5 minutes.
More informationIr O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )
Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O
More informationLINEAR ALGEBRA SUMMARY SHEET.
LINEAR ALGEBRA SUMMARY SHEET RADON ROSBOROUGH https://intuitiveexplanationscom/linear-algebra-summary-sheet/ This document is a concise collection of many of the important theorems of linear algebra, organized
More informationDefinition (T -invariant subspace) Example. Example
Eigenvalues, Eigenvectors, Similarity, and Diagonalization We now turn our attention to linear transformations of the form T : V V. To better understand the effect of T on the vector space V, we begin
More informationMath 118, Fall 2014 Final Exam
Math 8, Fall 4 Final Exam True or false Please circle your choice; no explanation is necessary True There is a linear transformation T such that T e ) = e and T e ) = e Solution Since T is linear, if T
More informationMATH 115A: SAMPLE FINAL SOLUTIONS
MATH A: SAMPLE FINAL SOLUTIONS JOE HUGHES. Let V be the set of all functions f : R R such that f( x) = f(x) for all x R. Show that V is a vector space over R under the usual addition and scalar multiplication
More information1. Diagonalize the matrix A if possible, that is, find an invertible matrix P and a diagonal
. Diagonalize the matrix A if possible, that is, find an invertible matrix P and a diagonal 3 9 matrix D such that A = P DP, for A =. 3 4 3 (a) P = 4, D =. 3 (b) P = 4, D =. (c) P = 4 8 4, D =. 3 (d) P
More informationMathematical foundations - linear algebra
Mathematical foundations - linear algebra Andrea Passerini passerini@disi.unitn.it Machine Learning Vector space Definition (over reals) A set X is called a vector space over IR if addition and scalar
More informationMATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION
MATH (LINEAR ALGEBRA ) FINAL EXAM FALL SOLUTIONS TO PRACTICE VERSION Problem (a) For each matrix below (i) find a basis for its column space (ii) find a basis for its row space (iii) determine whether
More informationMath 314/ Exam 2 Blue Exam Solutions December 4, 2008 Instructor: Dr. S. Cooper. Name:
Math 34/84 - Exam Blue Exam Solutions December 4, 8 Instructor: Dr. S. Cooper Name: Read each question carefully. Be sure to show all of your work and not just your final conclusion. You may not use your
More informationMath 113 Final Exam: Solutions
Math 113 Final Exam: Solutions Thursday, June 11, 2013, 3.30-6.30pm. 1. (25 points total) Let P 2 (R) denote the real vector space of polynomials of degree 2. Consider the following inner product on P
More informationRemark 1 By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.
Sec 5 Eigenvectors and Eigenvalues In this chapter, vector means column vector Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More informationMath 108b: Notes on the Spectral Theorem
Math 108b: Notes on the Spectral Theorem From section 6.3, we know that every linear operator T on a finite dimensional inner product space V has an adjoint. (T is defined as the unique linear operator
More informationMath 20F Final Exam(ver. c)
Name: Solutions Student ID No.: Discussion Section: Math F Final Exam(ver. c) Winter 6 Problem Score /48 /6 /7 4 /4 5 /4 6 /4 7 /7 otal / . (48 Points.) he following are rue/false questions. For this problem
More informationSUMMARY OF MATH 1600
SUMMARY OF MATH 1600 Note: The following list is intended as a study guide for the final exam. It is a continuation of the study guide for the midterm. It does not claim to be a comprehensive list. You
More informationMath 321 Final Exam. 1. Do NOT write your answers on these sheets. Nothing written on the test papers will be graded.
Math 31 Final Exam Instructions 1 Do NOT write your answers on these sheets Nothing written on the test papers will be graded Please begin each section of questions on a new sheet of paper 3 Do not write
More informationQ1 Q2 Q3 Q4 Tot Letr Xtra
Mathematics 54.1 Final Exam, 12 May 2011 180 minutes, 90 points NAME: ID: GSI: INSTRUCTIONS: You must justify your answers, except when told otherwise. All the work for a question should be on the respective
More informationLinear Algebra Review. Vectors
Linear Algebra Review 9/4/7 Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa (UCSD) Cogsci 8F Linear Algebra review Vectors
More informationMTH 5102 Linear Algebra Practice Final Exam April 26, 2016
Name (Last name, First name): MTH 5 Linear Algebra Practice Final Exam April 6, 6 Exam Instructions: You have hours to complete the exam. There are a total of 9 problems. You must show your work and write
More information3. Identify and find the general solution of each of the following first order differential equations.
Final Exam MATH 33, Sample Questions. Fall 6. y = Cx 3 3 is the general solution of a differential equation. Find the equation. Answer: y = 3y + 9 xy. y = C x + C is the general solution of a differential
More informationMath Linear Algebra II. 1. Inner Products and Norms
Math 342 - Linear Algebra II Notes 1. Inner Products and Norms One knows from a basic introduction to vectors in R n Math 254 at OSU) that the length of a vector x = x 1 x 2... x n ) T R n, denoted x,
More informationRemark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.
Sec 6 Eigenvalues and Eigenvectors Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called an eigenvalue of A if there is a nontrivial
More informationSTUDENT NAME: STUDENT SIGNATURE: STUDENT ID NUMBER: SECTION NUMBER RECITATION INSTRUCTOR:
MA262 FINAL EXAM SPRING 2016 MAY 2, 2016 TEST NUMBER 01 INSTRUCTIONS: 1. Do not open the exam booklet until you are instructed to do so. 2. Before you open the booklet fill in the information below and
More informationLinear Algebra in Actuarial Science: Slides to the lecture
Linear Algebra in Actuarial Science: Slides to the lecture Fall Semester 2010/2011 Linear Algebra is a Tool-Box Linear Equation Systems Discretization of differential equations: solving linear equations
More informationEigenvalues and Eigenvectors
/88 Chia-Ping Chen Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Eigenvalue Problem /88 Eigenvalue Equation By definition, the eigenvalue equation for matrix
More informationElementary linear algebra
Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The
More informationMATH 260 LINEAR ALGEBRA EXAM II Fall 2013 Instructions: The use of built-in functions of your calculator, such as det( ) or RREF, is prohibited.
MAH 60 LINEAR ALGEBRA EXAM II Fall 0 Instructions: he use of built-in functions of your calculator, such as det( ) or RREF, is prohibited ) For the matrix find: a) M and C b) M 4 and C 4 ) Evaluate the
More informationLinear Algebra Highlights
Linear Algebra Highlights Chapter 1 A linear equation in n variables is of the form a 1 x 1 + a 2 x 2 + + a n x n. We can have m equations in n variables, a system of linear equations, which we want to
More informationMathematical foundations - linear algebra
Mathematical foundations - linear algebra Andrea Passerini passerini@disi.unitn.it Machine Learning Vector space Definition (over reals) A set X is called a vector space over IR if addition and scalar
More informationMTH 2032 SemesterII
MTH 202 SemesterII 2010-11 Linear Algebra Worked Examples Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education December 28, 2011 ii Contents Table of Contents
More informationLinear Algebra- Final Exam Review
Linear Algebra- Final Exam Review. Let A be invertible. Show that, if v, v, v 3 are linearly independent vectors, so are Av, Av, Av 3. NOTE: It should be clear from your answer that you know the definition.
More information1. Let m 1 and n 1 be two natural numbers such that m > n. Which of the following is/are true?
. Let m and n be two natural numbers such that m > n. Which of the following is/are true? (i) A linear system of m equations in n variables is always consistent. (ii) A linear system of n equations in
More informationMATH 3321 Sample Questions for Exam 3. 3y y, C = Perform the indicated operations, if possible: (a) AC (b) AB (c) B + AC (d) CBA
MATH 33 Sample Questions for Exam 3. Find x and y so that x 4 3 5x 3y + y = 5 5. x = 3/7, y = 49/7. Let A = 3 4, B = 3 5, C = 3 Perform the indicated operations, if possible: a AC b AB c B + AC d CBA AB
More informationAssignment 1 Math 5341 Linear Algebra Review. Give complete answers to each of the following questions. Show all of your work.
Assignment 1 Math 5341 Linear Algebra Review Give complete answers to each of the following questions Show all of your work Note: You might struggle with some of these questions, either because it has
More informationMATHEMATICS COMPREHENSIVE EXAM: IN-CLASS COMPONENT
MATHEMATICS COMPREHENSIVE EXAM: IN-CLASS COMPONENT The following is the list of questions for the oral exam. At the same time, these questions represent all topics for the written exam. The procedure for
More information6 Inner Product Spaces
Lectures 16,17,18 6 Inner Product Spaces 6.1 Basic Definition Parallelogram law, the ability to measure angle between two vectors and in particular, the concept of perpendicularity make the euclidean space
More informationBASIC ALGORITHMS IN LINEAR ALGEBRA. Matrices and Applications of Gaussian Elimination. A 2 x. A T m x. A 1 x A T 1. A m x
BASIC ALGORITHMS IN LINEAR ALGEBRA STEVEN DALE CUTKOSKY Matrices and Applications of Gaussian Elimination Systems of Equations Suppose that A is an n n matrix with coefficents in a field F, and x = (x,,
More information1 9/5 Matrices, vectors, and their applications
1 9/5 Matrices, vectors, and their applications Algebra: study of objects and operations on them. Linear algebra: object: matrices and vectors. operations: addition, multiplication etc. Algorithms/Geometric
More informationHOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION)
HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION) PROFESSOR STEVEN MILLER: BROWN UNIVERSITY: SPRING 2007 1. CHAPTER 1: MATRICES AND GAUSSIAN ELIMINATION Page 9, # 3: Describe
More informationChapter 6 Inner product spaces
Chapter 6 Inner product spaces 6.1 Inner products and norms Definition 1 Let V be a vector space over F. An inner product on V is a function, : V V F such that the following conditions hold. x+z,y = x,y
More informationMIT Final Exam Solutions, Spring 2017
MIT 8.6 Final Exam Solutions, Spring 7 Problem : For some real matrix A, the following vectors form a basis for its column space and null space: C(A) = span,, N(A) = span,,. (a) What is the size m n of
More information18.06 Problem Set 8 - Solutions Due Wednesday, 14 November 2007 at 4 pm in
806 Problem Set 8 - Solutions Due Wednesday, 4 November 2007 at 4 pm in 2-06 08 03 Problem : 205+5+5+5 Consider the matrix A 02 07 a Check that A is a positive Markov matrix, and find its steady state
More informationMath 21b Final Exam Thursday, May 15, 2003 Solutions
Math 2b Final Exam Thursday, May 5, 2003 Solutions. (20 points) True or False. No justification is necessary, simply circle T or F for each statement. T F (a) If W is a subspace of R n and x is not in
More informationProperties of Linear Transformations from R n to R m
Properties of Linear Transformations from R n to R m MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Topic Overview Relationship between the properties of a matrix transformation
More informationPRACTICE PROBLEMS FOR THE FINAL
PRACTICE PROBLEMS FOR THE FINAL Here are a slew of practice problems for the final culled from old exams:. Let P be the vector space of polynomials of degree at most. Let B = {, (t ), t + t }. (a) Show
More informationMATH 369 Linear Algebra
Assignment # Problem # A father and his two sons are together 00 years old. The father is twice as old as his older son and 30 years older than his younger son. How old is each person? Problem # 2 Determine
More informationThe value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver I.N.
Math 410 Homework Problems In the following pages you will find all of the homework problems for the semester. Homework should be written out neatly and stapled and turned in at the beginning of class
More informationAnalysis Preliminary Exam Workshop: Hilbert Spaces
Analysis Preliminary Exam Workshop: Hilbert Spaces 1. Hilbert spaces A Hilbert space H is a complete real or complex inner product space. Consider complex Hilbert spaces for definiteness. If (, ) : H H
More information1 Linear Algebra Problems
Linear Algebra Problems. Let A be the conjugate transpose of the complex matrix A; i.e., A = A t : A is said to be Hermitian if A = A; real symmetric if A is real and A t = A; skew-hermitian if A = A and
More informationAnswer Keys For Math 225 Final Review Problem
Answer Keys For Math Final Review Problem () For each of the following maps T, Determine whether T is a linear transformation. If T is a linear transformation, determine whether T is -, onto and/or bijective.
More informationImplicit Functions, Curves and Surfaces
Chapter 11 Implicit Functions, Curves and Surfaces 11.1 Implicit Function Theorem Motivation. In many problems, objects or quantities of interest can only be described indirectly or implicitly. It is then
More informationHW2 Solutions
8.024 HW2 Solutions February 4, 200 Apostol.3 4,7,8 4. Show that for all x and y in real Euclidean space: (x, y) 0 x + y 2 x 2 + y 2 Proof. By definition of the norm and the linearity of the inner product,
More informationTangent spaces, normals and extrema
Chapter 3 Tangent spaces, normals and extrema If S is a surface in 3-space, with a point a S where S looks smooth, i.e., without any fold or cusp or self-crossing, we can intuitively define the tangent
More information4. Linear transformations as a vector space 17
4 Linear transformations as a vector space 17 d) 1 2 0 0 1 2 0 0 1 0 0 0 1 2 3 4 32 Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation
More informationPh.D. Katarína Bellová Page 1 Mathematics 1 (10-PHY-BIPMA1) EXAM SOLUTIONS, 20 February 2018
Ph.D. Katarína Bellová Page Mathematics 0-PHY-BIPMA) EXAM SOLUTIONS, 0 February 08 Problem [4 points]: For which positive integers n does the following inequality hold? n! 3 n Solution: Trying first few
More informationUniversity of Colorado at Denver Mathematics Department Applied Linear Algebra Preliminary Exam With Solutions 16 January 2009, 10:00 am 2:00 pm
University of Colorado at Denver Mathematics Department Applied Linear Algebra Preliminary Exam With Solutions 16 January 2009, 10:00 am 2:00 pm Name: The proctor will let you read the following conditions
More informationEigenvalues and Eigenvectors A =
Eigenvalues and Eigenvectors Definition 0 Let A R n n be an n n real matrix A number λ R is a real eigenvalue of A if there exists a nonzero vector v R n such that A v = λ v The vector v is called an eigenvector
More informationMATH 1553, C. JANKOWSKI MIDTERM 3
MATH 1553, C JANKOWSKI MIDTERM 3 Name GT Email @gatechedu Write your section number (E6-E9) here: Please read all instructions carefully before beginning Please leave your GT ID card on your desk until
More informationContents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2
Contents Preface for the Instructor xi Preface for the Student xv Acknowledgments xvii 1 Vector Spaces 1 1.A R n and C n 2 Complex Numbers 2 Lists 5 F n 6 Digression on Fields 10 Exercises 1.A 11 1.B Definition
More informationMa/CS 6b Class 23: Eigenvalues in Regular Graphs
Ma/CS 6b Class 3: Eigenvalues in Regular Graphs By Adam Sheffer Recall: The Spectrum of a Graph Consider a graph G = V, E and let A be the adjacency matrix of G. The eigenvalues of G are the eigenvalues
More informationDefinitions for Quizzes
Definitions for Quizzes Italicized text (or something close to it) will be given to you. Plain text is (an example of) what you should write as a definition. [Bracketed text will not be given, nor does
More informationMath 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.
Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,
More information